Soft robots are challenging to model and control due to their poorly defined kinematics and nonlinear dynamics. Recently, Koopman operator theory has been shown capable of constructing control-oriented soft robot models from data. However, building these models requires extensive data collection and they do not necessarily generalize well outside of the training observations. This paper presents a more data-efficient and generalizable approach to soft robot modeling that first identifies a physics-based Koopman model then supplements it with a data-driven residual Koopman model. The resulting combined model is linear and thus compatible with real-time model-based control techniques such as Model Predictive Control (MPC). The efficacy of the approach is demonstrated on several simulated systems and on a real soft robot arm, where it is shown to generate models that are more accurate than purely physics-based models and require less data to construct than purely data-driven models. Using a model-based controller, the soft arm is able to successfully track end effector trajectories, perform a pick-and-place task, and write on a dry-erase board, showcasing the applicability of this framework to increase the capabilities of soft robotic systems.
AbrahamIDe La TorreGMurpheyTD (2017) Model-based control using koopman operators. arXiv preprint arXiv1709: 01568.
2.
AsadaHH (2023) Global, unified representation of heterogenous robot dynamics using composition operators: a Koopman direct encoding method. IEEE28: 2633–2644. DOI: 10.1109/TMECH.2023.3253599. https://ieeexplore.ieee.org/document/10129822/
3.
AströmKJMurrayRM (2010) Feedback Systems: An Introduction for Scientists and Engineers. Princeton university press.
4.
BevandaPBeierMKerzS, et al. (2022) Diffeomorphically learning stable koopman operators. IEEE Control Systems Letters6: 3427–3432.
5.
BruderDWoodRJ (2022) The chain-link actuator: exploiting the bending stiffness of mckibben artificial muscles to achieve larger contraction ratios. IEEE Robotics and Automation Letters7(1): 542–548.
6.
BruderDSedalAVasudevanR, et al. (2018) Force generation by parallel combinations of fiber-reinforced fluid-driven actuators. IEEE Robotics and Automation Letters3(4): 3999–4006.
7.
BruderDGillespieBRemyCD, et al. (2019a) Modeling and control of soft robots using the Koopman Operator and model predictive control. In: Proceedings of Robotics: Science and Systems. Germany: FreiburgimBreisgau. DOI: 10.15607/RSS.2019.XV.060.
8.
BruderDRemyCDVasudevanR (2019b) Nonlinear system identification of soft robot dynamics using Koopman Operator theory. In: 2019 International Conference on Robotics and Automation (ICRA). Piscataway: IEEE, 6244–6250.
9.
BruderDFuXGillespieRB, et al. (2021) Data-driven control of soft robots using koopman operator theory. IEEE Transactions on Robotics37(3): 948–961.
10.
BruderDFuXGillespieRB, et al. (2021a) Koopman-based control of a soft continuum manipulator under variable loading conditions. IEEE Robotics and Automation Letters6(4): 6852–6859.
11.
BruderDFuXVasudevanR (2021b) Advantages of bilinear Koopman realizations for the modeling and control of systems with unknown dynamics. IEEE Robotics and Automation Letters6(3): 4369–4376. DOI: 10.1109/LRA.2021.3068117.
12.
BruntonSLBruntonBWProctorJL, et al. (2016) Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control. PLoS One11(2): e0150171.
13.
BudišićMMohrRMezićI (2012) Applied koopmanism. Applied Koopmanism Chaos: An Interdisciplinary Journal of Nonlinear Science22(4): 047510.
14.
ChenJDangYHanJ (2022) Offset-free model predictive control of a soft manipulator using the Koopman Operator. Mechatronics86: 102871.
15.
DahdahSForbesJR (2022) System norm regularization methods for koopman operator approximation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences478(2265): 20220162.
16.
Della SantinaCDuriezCRusD (2023) Model-based control of soft robots: a survey of the state of the art and open challenges. IEEE Control Systems43(3): 30–65.
17.
DernerEKubalikJBabuskaR (2021) Guiding Robot Model Construction with Prior Features. Piscataway: Institute of Electrical and Electronics Engineers Inc, 7112–7118. DOI: 10.1109/IROS51168.2021.9635831.
18.
EbersMRSteeleKMKutzJN (2022) Discrepancy modeling framework: learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects. arXiv preprint23(1): 440–469. https://arxiv.org/abs/2203.05164
19.
FlaccoFPaolilloAKheddarA (2016) Residual-based contacts estimation for humanoid robots. In: 2016 IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids). Piscataway: IEEE, 409–415.
20.
GazCMagriniEDe LucaA (2018) A model-based residual approach for human-robot collaboration during manual polishing operations. Mechatronics55: 234–247. DOI: 10.1016/j.mechatronics.2018.02.014.
21.
George ThuruthelTAnsariYFaloticoE, et al. (2018) Control strategies for soft robotic manipulators: a survey. Soft Robotics5(2): 149–163.
22.
GoswamiDPaleyDA (2020) Global bilinearization and reachability analysis of control-affine nonlinear systemsThe Koopman Operator in Systems and Control. Berlin: Springer, 81–98.
23.
GoyalTHussainSMartinez-MarroquinE, et al. (2023) Learning Koopman embedding subspaces for system identification and optimal control of a wrist rehabilitation robot. IEEE Transactions on Industrial Electronics70(7): 7092–7101.
24.
HaggertyDABanksMJCurtisPC, et al. (2020) Modeling, reduction, and control of a helically actuated inertial soft robotic arm via the Koopman Operator. arXiv preprint arXiv2011: 07939.
25.
HaggertyDABanksMJKamenarE, et al. (2023) Control of soft robots with inertial dynamics. Science Robotics8(81): eadd6864.
26.
HanYHaoWVaidyaU (2020) Deep Learning of Koopman Representation for Control. Piscataway: 2020 59th IEEE Conference on Decision and Control (CDC). IEEE, 1890–1895.
27.
HanMEuler-RolleJKatzschmannRK (2021) Desko: stability-assured robust control with a deep stochastic Koopman Operator. In: International Conference on Learning Representations. Vienna Austria: ICLR.
28.
HanMWongKEuler-RolleJ, et al. (2023) Robust learning-based control for uncertain nonlinear systems with validation on a soft robot. Piscataway: IEEE Transactions on Neural Networks and Learning Systems.
29.
HaraKInoueMSebeN (2020) Learning koopman operator under dissipativity constraints. IFAC-PapersOnLine53(2): 1169–1174.
30.
HighamNJ (2008) Functions of matrices: theory and computation. New Delhi: Siam, Vol. 104.
31.
JohanninkTBahlSNairA, et al. (2019) Residual reinforcement learning for robot control. In: 2019 International Conference on Robotics and Automation (ICRA). Piscataway: IEEE, 6023–6029.
32.
KahemanKKaiserEStromB, et al. (2019) Learning discrepancy models from experimental data. Ithaca: arXiv preprint. https://arxiv.org/abs/1909.08574
33.
KamenarEĆrnjarić-ŽicNHaggertyD, et al. (2020) Prediction of the behavior of a pneumatic soft robot based on Koopman Operator theory. In: 2020 43rd International Convention on Information, Communication and Electronic Technology (MIPRO). Piscataway: IEEE, 1169–1173.
34.
KanamaruT (2007) Van der pol oscillator. Scholarpedia2(1): 2202.
35.
KimJSQuanYSChungCC (2023) Koopman operator-based model identification and control for automated driving vehicle. International Journal of Control, Automation and Systems21(8): 2431–2443.
36.
KordaMMezićI (2018a) Linear predictors for nonlinear dynamical systems: koopman Operator meets model predictive control. Automatica93: 149–160.
37.
KordaMMezićI (2018b) On convergence of extended dynamic mode decomposition to the Koopman Operator. Journal of Nonlinear Science28(2): 687–710.
38.
LiQDietrichFBolltEM, et al. (2017) Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the koopman operator. Chaos27(10): 103111.
39.
LiSStampfliJJXuHJ, et al. (2019) A vacuum-driven origami “magic-ball” soft gripper. In: 2019 International Conference on Robotics and Automation (ICRA). Piscataway: IEEE, 7401–7408.
40.
LipsonH (2014) Challenges and opportunities for design, simulation, and fabrication of soft robots. Soft Robotics1(1): 21–27.
41.
LuschBKutzJNBruntonSL (2018) Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications9(1): 4950.
42.
LynchKMParkFC (2017) Modern Robotics. Cambridge: Cambridge University Press.
43.
MabrokMAAksikasIMeskinN (2023) Koopman operator approximation under negative imaginary constraints. Piscataway: IEEE Control Systems Letters.
44.
MamakoukasGCastanoMTanX, et al. (2019) Local Koopman Operators for data-driven control of robotic systems. In: Robotics: Science and Systems, Freiburg, Germany, June 22-26, 2019.
45.
MamakoukasGAbrahamIMurpheyTD (2023) Learning stable models for prediction and control. Piscataway: IEEE Transactions on Robotics.
MezićI (2013) Analysis of fluid flows via spectral properties of the koopman operator. Annual Review of Fluid Mechanics45: 357–378.
51.
NandanooriSPGuanSKunduS, et al. (2022) Graph neural network and koopman models for learning networked dynamics: a comparative study on power grid transients prediction. IEEE Access10: 32337–32349.
52.
NgJAsadaHH (2023) Data-driven encoding: a new numerical method for computation of the Koopman Operator. IEEE Robotics and Automation Letters8: 3940–3947. DOI: 10.1109/LRA.2023.3273515. https://ieeexplore.ieee.org/document/10120913/
53.
NozawaIKamienskiEO’NeillC, et al. (2024) A Monte Carlo approach to koopman direct encoding and its application to the learning of neural-network observables. IEEE Robotics and Automation Letters9(3): 2264–2271. DOI: 10.1109/LRA.2024.3354612. https://ieeexplore.ieee.org/document/10400968/
54.
OlverFW (2010) NIST Handbook of Mathematical Functions Hardback and CD-ROM. Cambridge: Cambridge University Press.
55.
RawlingsJBMayneDQ (2009) Model Predictive Control: Theory and Design. Madison, Wisconsin: Nob Hill Pub.
56.
RenCJiangHLiC, et al. (2023) Koopman-Operator-based robust data-driven control for wheeled mobile robots. IEEE28(1): 461–472.
57.
RousseeuwPJLeroyAM (2005) Robust regression and outlier detection. Hoboken: John Wiley & Sons, Vol. 589.
58.
RusDTolleyMT (2015) Design, fabrication and control of soft robots. Nature521(7553): 467–475.
59.
SelbyNSAsadaHH (2021) Learning of causal observable functions for koopman-dfl lifting linearization of nonlinear controlled systems and its application to excavation automation. IEEE Robotics and Automation Letters6(4): 6297–6304.
60.
SenanNAF (2007) A Brief Introduction to Using Ode45 in Matlab. Berkeley: University of California at Berkeley.
61.
ShiHMengMQH (2022) Deep koopman operator with control for nonlinear systems. IEEE Robotics and Automation Letters7(3): 7700–7707.
62.
ShiLMucchianiCKarydisK (2022) Online modeling and control of soft multi-fingered grippers via Koopman Operator theory. In: 2022 IEEE 18th International Conference on Automation Science and Engineering (CASE). Piscataway: IEEE, 1946–1952.
63.
ShiLLiuZKarydisK (2023) Koopman operators for modeling and control of soft robotics. Current Robotics Reports4(2): 23–31.
64.
StrangG (2012) Linear Algebra and its Applications. Pacific Grove: Thomson, Brooks/Cole.
65.
SusukiYMezicIRaakF, et al. (2016) Applied koopman operator theory for power systems technology. Nonlinear Theory and Its Applications, IEICE7(4): 430–459.
TibshiraniR (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society - Series B: Statistical Methodology58: 267–288.
WangHLiangWLiangB, et al. (2023) Robust position control of a continuum manipulator based on selective approach and Koopman Operator. Piscataway: IEEE Transactions on Industrial Electronics.
71.
WebsterRJIIIJonesBA (2010) Design and kinematic modeling of constant curvature continuum robots: a review. The International Journal of Robotics Research29(13): 1661–1683.
WilliamsMOKevrekidisIGRowleyCW (2015) A data–driven approximation of the Koopman Operator: extending dynamic mode decomposition. Journal of Nonlinear Science25(6): 1307–1346.
74.
YeungEKunduSHodasN (2019) Learning deep neural network representations for koopman operators of nonlinear dynamical systems. In: 2019 American Control Conference (ACC). Piscataway: IEEE, 4832–4839.
75.
YuSShenCErsalT (2022) Autonomous driving using linear model predictive control with a koopman operator based bilinear vehicle model. IFAC-PapersOnLine55(24): 254–259.
76.
ZhangJWangH (2023) Online model predictive control of robot manipulator with structured deep koopman model. IEEE Robotics and Automation Letters99: 1–8.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
